?Let \(X_{1}, X_{2}, \ldots, X_{n}\) be uniformly distributed on the interval 0 to a

Chapter 7, Problem 7.4.6

(choose chapter or problem)

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be uniformly distributed on the interval 0 to a. Recall that the maximum likelihood estimator of a is \(\hat{a}=\max \left(X_{i}\right)\).

a. Argue intuitively why \(\hat{a}\) cannot be an unbiased estimator for a.

b. Suppose that \(E(\hat{a})=n a /(n+1)\). Is it reasonable that \(\hat{a}\) consistently underestimates a? Show that the bias in the estimator approaches zero as n gets large.

c. Propose an unbiased estimator for a.

d. Let \(Y=\max \left(X_{i}\right)\). Use the fact that \(Y \leq y\) if and only if each \(X_{i} \leq y\) to derive the cumulative distribution function of Y. Then show that the probability density function of Y is

\(f(y)=\left\{\begin{array}{ll}\frac{n y^{n-1}}{a^{n}}, & 0 \leq y \leq a \\0, & \text { otherwise }\end{array}\right.\)

Use this result to show that the maximum likelihood estimator for a is biased.

e. We have two unbiased estimators for a: the moment estimator \(\hat{a}_{1}=2 \bar{X}\) and \(1 a_{2}=\lfloor(n+1) / n\rfloor \max \left(X_{i}\right)\), where \(max (X_{i})\) is the largest observation in a random sample of size n. It can be shown that \(V\left(\hat{a}_{1}\right)=a^{2} /(3 n)\) and that \(V(\hat{a}_{2})=a^{2} /[n(n+2)]\). Show that if \(n>1, \hat{a}_{2}\) is a better estimator than \(\hat{a}\). In what sense is it a better estimator of a?

Text Transcription:

X_{1}, X_{2}, \ldots, X_{n}

\hat{a}=max (X_{i})

\hat{a}

E(\hat{a})=n a /(n+1)

Y=max (X_{i})

Y \leq y

f(y)=(^\frac{n y^{n-1}}{a^{n}}, 0 \leq y \leq a _0 { otherwise }

X_{i} \leq y

\hat{a}_{1}=2 \bar{X}

a_{2}=[(n+1) / n] max (X_{i})

max (X_{i})

V(\hat{a}_{1})=a^{2} /(3 n)

V(\hat{a}_{2})=a^{2} /[n(n+2)]